Constructive aspects of wreath products and quasiprimitive permutation groups

• Konstruktive Aspekte von Kranzprodukten und quasiprimitiven Permutationsgruppen

Bernhardt, Dominik Hans; Niemeyer, Alice Catherine (Thesis advisor); Horn, Max (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

In 1993, Praeger introduced quasiprimitive permutation groups. A finite group acting on a finite set is said to act quasiprimitively if every non-trivial normal subgroup acts transitively. In a theorem similar to the famous O’Nan-Scott-theorem for primitive groups, Praeger classified quasiprimitive groups by dividing them into several mutually exclusive classes. Quasiprimitive permutation groups feature prominently in both the theory of permutation groups and in the study of symmetry groups of incidence structures. Yet, in contrast to arbitrary transitive or primitive permutation groups, no complete database of quasiprimitive groups up to a certain degree is available. A main result of this thesis is the construction of a database of all quasiprimitive but imprimitive permutation groups - called quimp groups - of degree at most 4095. Together with the database of primitive groups of degree at most 4095 constructed with contributions by many authors, a database of all quasiprimitive permutation groups of degree at most 4095 is now available. Praeger and Baddeley refined the original classification and divided quasiprimitive groups into eight mutually exclusive classes. To construct all quimp groups of degree at most 4095, we first observe that three of the eight types of quasiprimitive groups, namely groups of HA-, HS- and HC-type, are always primitive and that the minimal permutation degree of a quimp group of SD- or CD-type exceeds our degree bound. For the remaining three types of quasiprimitive groups, AS-type, PA-type and TW-type, we present structure theory and algorithms to construct such groups of a given degree. Our results are available in the GAP-package QuimpGrp which accompanies this thesis. Many (quasi-)primitive groups arise as subgroups of wreath products. The second main result of this thesis is a constructive description of the conjugacy classes and centralisers and the solution to the conjugacy problem for wreath products where the base group is any group and the top group acts faithfully on a finite set. Our approach is inspired by ideas originally developed by Ore, Specht, James and Kerber and our theory is presented in a way that is close to the implementation. Our results are implemented in a GAP-package by Rober.